Problem: Find the gradient of $f(x, y) = \dfrac{1}{x^2 + y^2}$. $\nabla f = ($ $,$ $)$
The gradient of a scalar field is all its partial derivatives put together into a vector. For a 2D scalar field, this looks like $\nabla f = (f_x, f_y)$. Let's find $f_x$ and $f_y$. $\begin{aligned} f_x &= \dfrac{\partial}{\partial x} \left[ \dfrac{1}{x^2 + y^2} \right] \\ \\ &= \dfrac{-2x}{(x^2 + y^2)^2} \\ \\ f_y &= \dfrac{\partial}{\partial y} \left[ \dfrac{1}{x^2 + y^2} \right] \\ \\ &= \dfrac{-2y}{(x^2 + y^2)^2} \\ \\ \end{aligned}$ [How did you find those partial derivatives?] The gradient of $f$ is $\nabla f = \left( \dfrac{-2x}{(x^2 + y^2)^2}, \dfrac{-2y}{(x^2 + y^2)^2} \right)$.